A Forty-Segment Molecular Synchrotron

Z. Phys. Chem. 227 (2013) 1605–1645 / DOI 10.1524/zpch.2013.0418© by Oldenbourg Wissenschaftsverlag, München

A Forty-Segment Molecular Synchrotron

By Peter C. Zieger1, Chris J. Eyles1, Sebastiaan Y. T. van de Meerakker2,Andre J. A. van Roij2, Hendrick L. Bethlem3, and Gerard Meijer1,2,∗1 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195 Berlin, Germany2 Radboud University Nijmegen, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ

Nijmegen, The Netherlands3 LaserLaB, Department of Physics and Astronomy, VU University Amsterdam, De Boelelaan 1081,

1081 HV Amsterdam, The Netherlands

We acknowledge the design of the electronics by G. Heyne, V. Platschkowski, andT. Vetter and the software support by U. Hoppe. C. J. E. thanks the Alexander vonHumboldt Foundation for a post-doctoral fellowship. S. Y. T. v. d. M acknowledgesfinancial support from NWO via a VIDI grant. H. L. B. acknowledges financialsupport from NWO via a VIDI-grant, and from the ERC via a Starting Grant. G. M.acknowledges support from the ERC-2009-AdG under grant agreement247142-MolChip.

(Received March 22, 2013; accepted in revised form July 18, 2013)

(Published online September 30, 2013)

Cooling and Trapping of Molecules / Synchrotrons / Molecular Collisions

We present a synchrotron for polar molecules consisting of 40 straight hexapoles arranged ina circle with a 50 cm diameter. The mechanical design and alignment procedure as well as thetrigger scheme used to switch the voltages applied to the hexapoles are described in detail. Thestability of the synchrotron is demonstrated by measurements in which multiple packets are storedfor over 13 s, during which they have completed over 1000 round trips and traveled a distance ofover one mile. Furthermore, we demonstrate the simultaneous trapping of 26 packets; 13 revolvingclock-wise and 13 counter clock-wise, that are injected into the synchrotron by two Stark deceleratorbeamlines. We discuss the opportunities for using the synchrotron as a low-energy collider.

1. IntroductionThe ability to control the motion of neutral molecules offers exciting new possibili-ties for spectroscopic and collision studies [1–3]. One method by which this can beachieved is to exploit the interaction of polar molecules with electric fields. A polarmolecule is a molecule whose charge is unequally distributed over the molecule, i. e.,one side of the molecule is more positively charged and one side is more negativelycharged. When such a molecule is placed in an electric field, the positive side of the

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1606 P. C. Zieger et al.

molecule experiences a force along the direction of the electric field, while the negativeside of the molecule experiences a force in the opposite direction to the electric field.If the electric field is homogeneous, the two forces cancel and no net force acts on themolecule. However, if the electric field is inhomogeneous, the two forces no longer can-cel. In this case, the net resultant force is given by: �F = μeff

�∇| �E| with μeff being theeffective dipole moment, which is given by the projection of the dipole moment ontothe electric field vector. It is instructive to compare this formula with Coulomb’s law,�F = −q �∇V , which describes the force exerted on a charged particle in a homogeneous

electric field. The similarity between the two formulas suggests that we can manipulatepolar molecules with electric field gradients in much the same way as charged particlesare manipulated with voltage differences. There are complications, however. First, μeff

depends on the quantum state, and so it is crucial that the molecules remain in the samequantum state throughout the experiment [4–6]. Second, in general μeff is a function ofthe electric field. Third, in contrast to the electrostatic potential, the gradient of the elec-tric field magnitude in free space is not equal to zero [7]. The second and third pointintroduce aggravating non-linearities in electric lenses [8,9]. Last but not least, the forceexperienced by polar molecules in inhomogeneous fields is typically some eight to tenorders of magnitude weaker than the forces that can be exerted on ions. Nevertheless,over the last decade it has been demonstrated that the motion of molecules in a mo-lecular beam can be completely controlled using electric fields (see Ref. [10,11] andreferences therein).

This paper describes a synchrotron for polar molecules. It demonstrates the levelof control that can nowadays be achieved over molecules; multiple packets of neu-tral molecules are made to revolve clockwise and anti-clockwise for more than 1000revolutions in a tabletop storage ring, thereby traveling a distance of over one mile.

Confining molecules in a storage ring. Modern synchrotrons for charged particles aretypically built out of many individual elements placed along a circular track [12,13].These elements, separated by some free-flight distance, are designed for performinga specific task, e. g., injecting, steering, focusing, accelerating, bunching or extractingthe particles. A scheme for building a synchrotron for molecules in a similar fashionwas presented by Nishimura et al. [14], however, the acceptance of such a design isseriously hindered by the non-linearities of the forces experienced by the molecules.We have opted for a rather different approach. In our synchrotron, only one element –a hexapole – is used. It can be shown that the electric field around the average radialposition of the molecules (the equilibrium orbit, see Sect. 2.3), can be decomposed intoa constant, a linear and a quadratic term [15]. The linear term delivers the necessaryforce for steering the beam, the quadratic term focuses the molecule transversally, whileby modulating the constant term the molecules are focused in the longitudinal direction.To inject and extract the molecules the voltages on the electrodes are adjusted to obtaina homogeneous electric field of relatively small magnitude. Although, this was not ob-vious to us from the outset, the analysis presented in Ref. [15] suggests that a hexapolegeometry, approximated by circular rods, is in fact the most favorable geometry in termsof acceptance, ease of construction and required electronics.

In the experiments presented in this paper, we deal exclusively with polar moleculesin low-field seeking states, i. e., molecules that have their dipole moment oriented anti-



A Forty-Segment Molecular Synchrotron 1607

Fig. 1. The two predecessors of the present molecular synchrotron. The top left panel shows a photo real-istic image of the first generation storage ring for neutral molecules, consisting of a hexapole guide bentinto a circle of radius 125 mm. Below the ring, the ND3 intensity at the detection point is shown as a func-tion of storage time (reproduced from [23]). To guide the eye, the dashed line indicates a 1/t decay. Theinset shows the ammonia intensity for the 49th to the 51st round trips. It is seen that after 50 round trips thepacket has filled the entire ring.The top right panel shows the second generation molecular storage ring: the prototype molecular syn-chrotron, consisting of two hexapoles bent into a semi circle, separated by 2 mm wide gaps. The measure-ment under the ring shows the ND3 intensity as a function of storage time (reproduced from [24]). Here,due to the applied switching scheme, the molecular packet is confined as a tight bunch in an effective threedimensional potential well.

parallel to the electric field. Schemes for synchrotrons that store molecules in high-fieldseeking states based on alternating gradient focusing have been proposed by Nishimuraet al. [16] and by de Nijs and Bethlem [17], while a storage ring composed of a toroidalwire was proposed by Loesch and Scheel [18]. Magnetic storage rings have been usedto store low energy neutrons [19] and atoms [20–22].

The molecular storage ring described in this paper is the third generation storagering for polar molecules built in our laboratory. The first and second generation storagerings are shown schematically in Fig. 1. The top left panel of Fig. 1 shows a storage ringthat is obtained by bending a linear hexapole focuser into the shape of a torus, as pro-posed by Katz [25] and demonstrated by F. M. H. Crompvoets and coworkers [26]. Inthis prototype storage ring, the molecules are transversely confined but are essentiallyfree along the longitudinal direction, i. e., along the circle. The lower left panel of Fig. 1shows a measurement of the ND3 density at the detection point in the ring as a func-tion of storage time. For long storage times the spatial width of the packet graduallyincreases until, eventually, the molecules fill the entire ring.

After the successful implementation of this first storage ring, C. E. Heiner andcoworkers demonstrated a sectioned storage ring consisting of two hexapoles bent intoa semicircle with a radius of 125 mm, separated by a 2 mm gap [24]. By switchingthe electric field synchronously with the motion of the molecules, the electric fieldsin the gap are used to keep the molecular packet confined in the longitudinal direc-




tion. A photo realistic image of this molecular synchrotron is shown in the top rightpanel of Fig. 1. The lower right panel of Fig. 1 shows a measurement of the ND3

density as a function of storage time at the detection point. By switching the elec-tric fields, the width of the packet after 40 round trips was confined to only 3 mm. Inlater measurements, Heiner and coworkers demonstrated that a packet of molecules waskept in a tight bunch even after completing 100 round trips, corresponding to a flightdistance of 80 m [27]. The same group also showed the successful demonstration ofthe simultaneous confinement of two molecular packets, trailing each other by about200 mm [24].

This paper describes the third generation of a molecular storage ring, consist-ing of forty straight hexapoles [28]. The increased number of segments has severaladvantages:

(i) the transverse acceptance of the synchrotron increases when the symmetry of thesynchrotron increases. Variation of the confinement force as a function of the position inthe ring leads to parametric amplification of the amplitude of the motion inside the ringat particular longitudinal velocities. These motional resonances are absent if the period-icity of the ring is much smaller than the typical length scale of the oscillatory motion.Note that motional resonances are also introduced by imperfections in the constructionof the ring. However, if these imperfections are not too large the width and depth ofthese resonances are small and losses are kept to a minimum by carefully choosing thebest operation conditions (see Sect. 3.4).

(ii) The depth of the longitudinal potential well scales with the inverse of the lengthof a single segment. Compared to the synchrotron that consisted of two half rings, thelongitudinal well depth is expected to increase by a factor

√10. Actually, in this paper

a different bunching scheme is implemented that is a factor of three less efficient butmuch less demanding for the high voltage electronics. As a result the longitudinal welldepth is about the same as that of Heiner et al.; however, we have come to realize thatthe previously used bunching scheme led to additional losses from the ring due to insta-bilities of the molecular trajectories and nonadiabatic transitions to nontrapped states.In the new bunching scheme these losses are absent.

(iii) The number of packets that can be trapped scales with the number of seg-ments. In the experiments presented in this paper, we have trapped up to 13 packets ofmolecules revolving clock and counter-clock wise in the ring.

Although the synchrotron is useful for many studies, this paper will mainly focuson its possible use in collision studies. The ability to control the translational energyand energy spread of a molecular beam enables scattering experiments to be conductedthat probe molecular interaction potentials in great detail. A holy grail of this researchis to be able to observe resonances of the collision cross section as a function of col-lision energy. These resonances occur when kinetic energy is converted into rotationalenergy as a result of the anisotropy of the potential energy surface (PES) of the two col-lision partners; the width of the resonances is determined by the lifetime of this collisioncomplex [29,30]. The velocity spread of a Stark-decelerated beam is in principle suf-ficiently small to resolve this resonant structure. Another motivation for research intocold collisions is the fact that at low temperatures the collision process becomes sensi-tive to externally applied electric or magnetic fields. This gives a handle to control andsteer the outcome of a chemical reaction [31,32].




In 2006, Gilijamse et al. performed the first scattering experiment using a Stark-decelerated molecular beam [33]. In this experiment, OH radicals were scattered witha conventional beam of Xe atoms in a crossed molecular beam geometry. The totalcollision energy is changed by varying the velocity of the OH radicals, while thevelocity of the rare gas beam is kept fixed. Subsequently, collisions of OH radicalswith other rare gas atoms have been studied extensively, determining accurately thethreshold behaviors of the state-to-state inelastic scattering cross sections [34,35]. Re-cently, experiments have been conducted using collisions between Stark-deceleratedOH molecules and a state-selected beam of NO molecules [36]. Ultimately, one wouldlike to study collisions between two velocity controlled molecular beams to resolvethese resonances. In order to achieve this, one aims for molecular beams that arefully controlled, i. e., in a single quantum state and with a controlled absolute velocityand velocity spread. Thus far, the densities achieved have been insufficient for suchexperiments.

The obvious advantage of a ring structure for collision studies is that a stored packetof molecules revolving around the ring will repeatedly encounter counter-propagatingpackets. By storing many packets over an extended time, the sensitivity with which col-lisions can be detected increases by orders of magnitude compared to an experimentwhere the collision partners encounter each other only once. In Sect. 4 measurementswill be presented demonstrating the simultaneous trapping of 13 packets revolving inboth directions. A packet that has completed 64 1

4round trips (see Fig. 21) will have had

1670 encounters with packets revolving in the opposite direction.

Outline of this paper. In this paper we describe our experiments with a 40-segmentsynchrotron for polar molecules. It is based on the PhD-thesis of P. C. Zieger. Sec-tion 2 derives the theoretical background necessary to describe and understand themotion of the molecules inside the molecular synchrotron. Expressions are given forthe trap frequencies that characterize the motion of the molecules inside the mov-ing potential well. The equations presented in Sect. 2 are partially reproduced fromthe PhD theses of F. M. H. Crompvoets [23] and C. E. Heiner [37]. Section 3 presentsexperiments that demonstrate that full control is achieved over the motion of thepacket of state-selected molecules in a molecular storage ring. A decelerated molecu-lar packet is confined for over 13 s; at this time it has traveled over a mile and haspassed through 41 000 gaps. The distance of one mile testifies to the intrinsic stabil-ity of the molecular trajectories. A number of other measurements that characterizethe synchrotron are also presented in this section. Finally, Sect. 4 shows the simul-taneous confinement of counter-propagating packets. The currently achievable densi-ties of the trapped molecules are not yet sufficient to detect the desired bi-molecularcollisions.

2. Confining neutral molecules in a synchrotron

Figure 2 shows a schematic of the molecular synchrotron. It consists of forty straighthexapoles which are placed on a circle with a diameter of half a meter (Rring = 250 mm).Neighboring hexapoles are separated by a 2 mm gap. Each hexapole segment has an in-




Fig. 2. A schematic of the molecular synchrotron. The top panel (A) shows a top view of the forty straighthexapoles aligned onto a circle with a radius of 250 mm. The angle between adjacent hexapoles is 9◦. TwoStark decelerator beam lines inject packets of molecules tangentially at two positions in the synchrotron.When a new packet is injected, the two hexapoles shown in green and the two hexapoles shown in redare switched off at the appropriate times. The hexapoles in red are also used as extraction field for thelaser-produced ions. The inset shows a detailed view of a hexapole segment. To ensure a constant gapof 2 mm the electrode rods have different lengths. The lower panel (B) shows the cross section insidea hexapole segment. Following Anderson [38], the ratio between the rod radius Rhex and the inner radiusof the hexapole r0 is chosen to be 0.565.

ner radius of r0 = 3.54 mm. The electrodes have a radius of Rhex = 2 mm, such that theratio of Rhex/r0 is 0.565, which yields a good approximation to an ideal hexapole fieldusing cylindrical rods [38]. In order to maintain a constant gap of 2 mm between twoneighboring hexapole segments, electrodes in each hexapole segment are longer on theoutside and shorter on the inside. The position of the molecule, r, is described with thethree coordinates φ, r ′ and y′. φ is the angle of the molecule in the x-z-plane relative tothe detection point of the first beam line (between the two red hexapole segments of thefirst beam line on the top panel of Fig. 2). r ′ is the radial coordinate and y′ the verticalcoordinate with respective to the center of the hexapole. More details on the experimen-tal setup are given in Sect. 3. In this section the equations of motion for polar moleculesinside the molecular synchrotron are derived.




Fig. 3. A simplified schematic of the different oscillation types that a molecule performes in the molecu-lar synchrotron. The black curve represents the trajectory of the synchronous molecule (equilibrium orbit).The red curve is a trajectory of a non-synchronous molecule performing various oscillations. The upperpart shows the angular cyclotron frequency Ωcycl, which is 2π over the time that the synchronous moleculesneeds to complete one full round trip. The synchrotron frequency ωsyn is the oscillation frequency in thelongitudinal direction. The lower panel shows the two transverse oscillations. On the left is the horizontalor radial betatron oscillation ωr and on the right the vertical betatron oscillation ωy.

2.1 Characteristic frequencies

Figure 3 shows a simplified schematic of the different types of oscillations performedby molecules inside the molecular synchrotron. The motion is best described by fourcharacteristic frequencies. In keeping with the nomenclature used for charged particleaccelerators, we refer to these as the cyclotron, synchrotron, and (horizontal and ver-tical) betatron frequencies. The black curves represent the trajectory of a hypotheticalmolecule that makes a closed orbit in the synchrotron. We call this molecule the syn-chronous molecule, while we call its orbit the equilibrium orbit. By definition, theequilibrium orbit is at the position where the centrifugal force and the force createdby the hexapoles cancel each other [39]. The cyclotron frequency is the inverse of thetime that the molecule needs to complete one full round trip (round-trip time). Thecyclotron angular frequency is Ωcycl = 2π/trt. In our experiments Ωcycl/2π is typically78 Hz. While the cyclotron frequency describes the velocity of the traveling potentialwell, molecules will oscillate in all three directions within this moving frame. In thelongitudinal direction the molecules oscillate at the synchrotron frequency, ωsyn. In thetransverse directions the molecules oscillate with the horizontal and vertical betatronfrequency, ωr (lower left panel) and ωy (lower right panel), respectively. The character-istic frequencies depend on the confining voltages that are applied on the electrodes aswell as on the velocity of the traveling potential well.

The characteristic frequencies are derived in the following sections and aresummarized in Table 1 for a fixed velocity and confining voltage difference. In




Table 1. Characteristic frequencies of ammonia molecules in a forty-segment molecular synchrotron. Thelongitudinal velocity of the stored packet is 124.3 m/s while a voltage difference of 6 kV is used. The firstcolumn shows the theoretical values that are obtained in Sections 2.3 and 2.5. The middle column liststhe experimentally determined frequencies presented in this paper while the last column shows the resultsobtained via the resonant excitation technique [40].

type frequencies

theoretical experimental experimental(this section) (Section 3) (from Ref. [40])

cyclotron frequencyΩcycl/2π

78 Hz 78 Hz 78 Hz

synchrotron frequencyωsyn/2π

58 Hz 68 (2) Hz 64 (2) Hz

horizontal betatron frequencyωr/2π

879 Hz 891(11) Hz 890 (1) Hz

vertical betatron frequencyωy/2π

860 Hz 851 (4) Hz 851 (1) Hz

Sect. 3, both betatron frequencies are determined from the observed stop bandsin the synchrotron, while the synchrotron frequency is determined from expansionmeasurements. These frequencies are listed in the second column of Table 1. InZieger et al. [40], the characteristic frequencies are derived from measurements inwhich the amplitude, or the duration, of the HV-pulses that are applied to the hexapolesis modulated.

2.2 The Stark effect of the ammonia molecule

The ammonia molecule. The ammonia molecule is a classic textbook example ofa pyramid-shaped symmetric-top molecule [41]. The three hydrogen atoms (or in thecase of its deuterated form ND3 the three deuterium atoms) build the ground plane ofthe pyramid while the nitrogen atom sits at its apex. The rotational structure of NH3 andND3 is best described by two principal quantum numbers J and K , which are the totalangular momentum and its projection onto the molecular axis, respectively. The um-brella motion in ammonia is strongly hindered due to the interaction of the hydrogenatoms with the nitrogen that leads to a barrier. Due to quantum mechanical tunnelingthrough this barrier each rotational level in the ground state of ammonia is split intoa symmetric and an antisymmetric component. For the |J, K〉 = |1, 1〉 state, used in theexperiments described in this paper, this splitting is 0.79 cm−1 [42] and 0.053 cm−1 [43]for NH3 and ND3, respectively.

The Stark effect in ammonia. The charges of ammonia isotopologues are unevenlydistributed over the molecule, leading to a body-fixed electric dipole moment. In theground state of NH3 this dipole moment is equal to 1.47 Debye [44], while it is1.50 Debye in the ground state of ND3 [45]. The Stark shift is given by the eigenvalues




Fig. 4. The Stark energy of the |J, K〉 = |1, 1〉 state of NH3 and ND3 as a function of the electric field. Thered and the black arrows indicate the change in Stark energy at an electric field strength of 50 kV/cm forammonia and its deuterated isotopologue, respectively.

of H0 + HStark, where H0 is the field-free hamiltonian and HStark is given by

HStark = −μ ·E . (1)

It is convenient to use rigid rotor wave functions |J, K, M, p〉 as the eigenfunctions ofthe rotational state, where M is the projection of the total angular momentum on thespace-fixed axis, μ is the electric dipole moment and E the electric field, while p in-dicates the parity. The upper and lower inversion doublet of ammonia have oppositeparity, thus they are coupled by the Stark Hamiltonian in first order. Assuming that theenergy difference between different rotational states is large, the resulting Stark energyof the interacting inversion doublet levels is to a first approximation given by [45]

EStark = ±√(

Winv

2

)2

+(

μEMK

J(J +1)

)2

∓(

Winv

2

), (2)

with Winv being the above mentioned inversion splitting of ammonia.Figure 4 shows the energy of the ground state of para-ammonia as a function of

the electric field. The product of M and K , which in the following will be referred toas MK , is 0 or +1 in the symmetric state, and 0 or −1 in the antisymmetric state. Inthe presence of an electric field some energy levels shift with increasing electric field.For MK = +1 the Stark energy decreases with increasing electric field strength, whilefor MK = −1 the Stark energy increases. Molecules in the MK = −1 state are calledlow-field-seekers because they lower their energy by moving to regions with the low-est field. Conversely, molecules in the MK = +1 state are high-field-seekers. To a firstorder approximation, the MK = 0 states are not affected by the electric field.

The main reason why the experiments in this paper are performed with ND3 andnot with NH3 is due to the different Stark effect of both isotopologues. As indicated




by the red and black arrows in Fig. 4, the relative change in Stark energy of a low-field-seeking molecule in a given electric field is larger for ND3 than for NH3. In otherwords, a deuterated ammonia molecule which moves into an inhomogeneous electricfield will gain more Stark energy than its non deuterated isotopologue. It will thereforelose more kinetic energy which is advantageous for the process of Stark deceleration(see [46]). A second benefit of ND3 is the linearity of the Stark shift. Compared toNH3, ND3 has a small inversion splitting and is thus more sensitive to low electric fields(0–50 kV/cm).

2.3 Transversal confinement

Force, equilibrium radius and equilibrium height. In a perfect hexapole that is bentinto a torus – such as the first storage ring for neutral molecules that was built in ourlab [39] – the centrifugal force is balanced by the radial component of the hexapolefield. The centrifugal force can be written as

Fcentri =mv2

φ

Rring + r ′ . (3)

Here, r ′ is the radial position inside the hexapole, vφ is the longitudinal velocity and mthe mass of the molecule inside the synchrotron. The force in the radial direction ona deuterated ammonia molecule in the low-field seeking |J, MK〉 = |1,−1〉 state insidea hexapole is given by

Fhex = − kr ′√1+

(Winv

k(r′2+y′2)

)2. (4)

Here, Winv is the inversion splitting of this state and k is the harmonic force constantwhich is itself determined by the voltage difference V0 that is applied to the hexapolerods, the inner radius r0 of the hexapole as well as by the quantum state of the molecule

k = μ3V0

2r30

|MK |J(J +1)

. (5)

Combining both equations leads to

mv2φ

Rring + r ′ = kr ′√1+

(Winv

k(r′2+y′2)

)2. (6)

The particular radius where the two forces cancel each other is referred to as the equi-librium radius, r ′

equi. The inversion splitting in deuterated ammonia is small comparedto kr ′2

equi and Eq. (6) can be rewritten as

r ′equi =

Rring

2

⎡⎣√1+

(2vφ

Rringω

)2

−1

⎤⎦ ≈ v2φ

Rringω2

(7)




Fig. 5. Left panel: The Stark energy WStark for an ND3 molecule in the low-field seeking |J, MK〉 =|1,−1〉 state as a function of displacement from the hexapole center (blue curve) (y′ = 0 mm). With a for-ward velocity of 124.3 m/s the molecule finds itself inside the molecular synchrotron in a centrifugalpseudo-potential Wcentri (green curve). The sum of the two energies is the effective potential energy Wtotal

(red curve) that the molecules experience transversally (Wtotal = WStark + Wcentri). Right panel: cross sectionof one hexapole segment showing the electric field.

with ω, the betatron frequency given by

ω =√

k

m. (8)

The last approximation in Eq. (7) is valid if 2vφ � Rringω. In the vertical direction theforce of gravity needs to be compensated,

−mg = ky′√1+

(Winv

k(r′2equi+y′2)

)2. (9)

In analogy to the equilibrium radius, we will refer to the particular height where thetwo forces cancel each other as the equilibrium height, y′

equi. If we again neglect theinversion splitting we find

y′equi ≈ − g

ω2. (10)

When a voltage difference of 6 kV is applied between the rods of the hexapole, the beta-tron frequency, ω/(2π) for deuterated ammonia in the low-field seeking |J, K〉 = |1, 1〉state is equal to 880 Hz (i. e., in a perfect hexapole ring, when the inversion splitting isneglected, the radial and vertical betatron frequencies are equal). At a forward velocityof 124.3 m/s, we find from Eqs. (7) and (10) an equilibrium radius of r ′

equi = 2.0 mm andan equilibrium height of y′

equi = −0.32 μm. The small gravitational distortion justifiesthe neglect of the effect of gravity hereafter.

The above derivation assumed a perfect hexapole field. The right panel of Fig. 5shows the electric field for a hexapole composed of cylindrical rods. As a result of usingcylindrical rods, the electric field close to the electrodes is seen to deviate from theideal field. If this effect, as well as the inversion splitting, are taken into account theequilibrium radius becomes 2.1 mm.




Potential energy and betatron frequency. The effective potential energy Wtotal thata molecule experiences in the molecular synchrotron is the sum of the Stark energyWStark and the pseudo-potential energy of the centrifugal force, Wcentri . By integrating theleft side of Eq. (6) over dr ′, the pseudo-potential can be written as

Wcentri = −mv2φ ln

∣∣∣∣1+ r ′

Rring

∣∣∣∣ . (11)

Neglecting the inversion splitting, the resulting potential is

Wtotal = 1

2kr ′2 −mv2

φln

∣∣∣∣1+ r ′

Rring

∣∣∣∣ . (12)

The left panel of Fig. 5 shows the Stark energy (blue curve), centrifugal psuedo-potential (green curve) and the sum of these two potentials (red curve) as a function ofthe radial position in the ring. These calculations use the numerically calculated elec-tric field and include the inversion splitting. The forward velocity is taken as 124.3 m/s.With increasing velocity, the tilt of the pseudo-potential becomes steeper and the effect-ive total potential energy becomes shallower while the minimum of the total potentialenergy (the equilibrium radius) is displaced further from the geometrical center of thehexapole. In Sect. 3.4, measurements are presented where the velocity of the moleculesis varied. In these measurements, the attractive potential well is kept at constant depthand position by adjusting the voltages applied to the hexapole electrodes in accordancewith the velocity of the molecular packet.

The radial betatron frequency can be found by taking the second derivative of the ef-fective potential energy with respect to the radial position at the equilibrium radius. Fora longitudinal velocity of 124.3 m/s we find ωr/2π = 879 Hz. Similarly, the verticalbetatron frequency is found to be ωy/2π = 860 Hz.

Phase-space acceptance. An important quantity that characterizes the synchrotron isthe phase-space acceptance, A. The phase-space acceptance is defined as the volumein position and velocity space that can be occupied by the molecules, that is, a num-ber with units of (m ·m/s)3. Depending on the context the acceptance may also referto the boundary that encloses the phase-space volume occupied by the molecules [11].Assuming a linear restoring force, the acceptance of a ring is

Ar = πrmaxvr,max (13)

with rmax and vr,max being the distance to the equilibrium radius and the radial velocitya molecule may have while still being confined in the ring. Assuming an ideal hexapoleand neglecting the inversion splitting, the maximum distance to the equilibrium radiusis

rmax = r0 − r ′equi . (14)

The corresponding maximum transverse velocity vr is

vr,max = ωrmax ≈ ω

(r0 − v2

φ

Rringω2

). (15)




Fig. 6. The figure shows the maximum transverse velocity a molecule can have and still be confinedtransversally in the synchrotron for three different situations: The blue curve corresponds to calculations inwhich the inversion splitting is taken into account and the hexapole field is calculated with a finite-elementprogram [47]. For the green curve the inversion splitting is set to zero. The red curve corresponds to cal-culations that use the electric field for an ideal hexapole and neglects the inversion splitting (ω/(2π) =879 Hz). The shaded region shows the velocity range normally used in the molecular synchrotron.

With a forward velocity of 124.3 m/s and a voltage difference of 6 kV a maximumtransverse velocity of 8.4 m/s is accepted in the synchrotron. The radial acceptance is38 mm ·m/s. Figure 6 shows the maximum transverse velocity vr,max as a function offorward velocity vφ. The red curve assumes a perfect hexapole and is calculated usingEq. (15) at a constant frequency of 879 Hz. The green and blue curve show the max-imum velocity obtained from the effective potential using the numerically calculatedelectric field and setting the inversion splitting at zero (green curve) or at its correctvalue (blue curve).

By combining Eq. (13) to (15) we find

Ar = πω

(r0 − v2

f

Rringω2

)2

. (16)

In a similar fashion, the vertical acceptance of the ring is given by

Ay = πymaxvy,max = πωy2max = πω(r2

0 − r ′2equi) = πω

(r2

0 −(

v2f

Rringω2

)2)

. (17)

For a longitudinal velocity of 124.3 m/s the maximum vertical velocity is 15.3 m/s andthe acceptance is 120 mm ·m/s.

2.4 Transverse stability in a segmented ring

All equations derived in the previous sections are valid for a hexapole that is bent intoa perfect circle. In our synchrotron, we use forty straight hexapoles to approximate a cir-cle. In this section, we analyse how this changes the acceptance and motivate why wehave decided to use forty hexapoles and not more or less.1 To understand this number

1 This question is addressed in the thesis of C. E. Heiner [37]. Her analysis is partly reproduced here.




we need to look at the occurrence of so-called motional resonances. Imperfections ina circular structure lead to transverse loss as the molecules experience a periodic de-viation in the restoring force that leads to parametric amplification of the amplitude ofthe oscillatory motion of the molecules in the ring. This phenomenon is well known inparticle accelerators [13] and has already been investigated in the previous molecularsynchrotron [48]. Motional resonances occur if the number of transverse oscillationsper segment is either a half integer or whole integer value. The radial and vertical be-tatron tune νr and νy are defined as the number of transverse oscillations per segment:

νr = ωr

nsegωcycl

and νy = ωy

nsegωcycl

, (18)

with nseg the number of (identical) segments that the ring is composed of. Primarily,ωcycl depends on the velocity of the molecular packet and ωr,y on the applied voltage.As a result of motional resonances the trajectories are unstable for certain combinationsof the velocity and applied voltage. In keeping with the terminology used in particlephysics the unstable regions are called ‘stop bands’ [13]. Under the assumption that themolecule experiences a linear Stark effect inside a perfect hexapole field, the betatrontune in the radial and vertical direction is given by

νr,y =√

kr,y/m

nseg2πv f /(Rring + requi

) = Rring + requi

nsegvf

√3μeff V

mr30

. (19)

Forty segments make almost a good circle. The upper panel in Fig. 7 shows theacceptance that results from simulating the trajectories of 20 000 molecules througha molecular synchrotron consisting of different number of linear hexapoles as a func-tion of the forward velocity. The radius of the molecular synchrotron is 0.25 m andthe hexapoles have an inner radius r0 of 3.54 mm. The force on a molecule insidea hexapole segment n is assumed to be perfectly linear with a fixed frequency of 879 Hzand a force constant of

k = mω2 . (20)

The position of a molecule inside the molecular synchrotron in the x-z-plane is de-scribed with the angle φ which necessitates a coordinate transformation to the labora-tory frame (see Fig. 2)

sin φ = z

r= z√

x2 + z2. (21)

A molecule is in the straight hexapole segment n, if the following condition is fulfilled

n −1/2

nseg

<φ

2π<

n +1/2

nseg

, (22)

This model assumes that there are no gaps between individual straight parts. A moleculeis considered to be lost if its distance to the geometrical center of a hexapole becomes




Fig. 7. Upper panel: The numerically calculated phase-space acceptance in the horizontal plane of a mo-lecular synchrotron as a function of forward velocity and number of straight segments. The radial fre-quency is set to ωr/(2π) = 879 Hz. The horizontal dashed white line corresponds to a synchrotron consist-ing of forty hexapoles. Lower panel: The phase-space acceptance for a synchrotron consisting of either 40hexapoles (red curve) or 100 hexapoles (green curve). The black curve shows the acceptance for an idealhexapole torus using Eq. (16).

larger than the inner radius of the hexapole

r0 <√

y′2 + r ′2 . (23)

The fraction of molecules that remain inside the molecular synchrotron after 200 ms ismultiplied by the initial phase-space distribution to obtain the phase-space acceptanceA. The initial position spread of the random packet is chosen to be 8 mm and the initialvelocity spread is 40 m/s which is larger than the acceptance for all settings. Each pointin the upper panel of Fig. 7 represents the one dimensional phase-space acceptance ata specific number of segments and velocity. It is obvious that when the ring is composedout of more straight segments, the motional resonances are shifted to lower velocitiesand become less dominant. In other words, the higher the symmetry of the synchrotron(number of straight segments) the fewer motional resonances occur. The black curve inthe lower panel of Fig. 7 shows the acceptance for a perfect ring symmetry and is calcu-lated using Eqs. (16) and (7). The green curve shows the simulated acceptance for 100segments. Only for velocities below 50 m/s does it differ from the ideal case. The redcurve shows the transverse stability for 40 segments. From this graph it was decidedto build the current forty segment molecular synchrotron as a compromise between ac-ceptance and construction effort. The calculation in Fig. 7 assumes that every hexapoleis perfectly aligned and the same voltage difference is applied to each hexapole seg-ment. In reality, the alignment of each hexapole segment with respect to the centerline




Fig. 8. Left panel: Scheme of the time dependence of the voltage difference between adjacent hexapoleelectrodes during operation of the synchrotron. By temporarily switching to a higher voltage differencewhenever the packet passes through a gap, ammonia molecules are kept in a tight bunch while revolving.The green dashed curve indicates the equilibrium orbit at a forward velocity of 124.3 m/s. Right panel:Electric field strength (both for confinement and bunching) as a function of position along the equilibriumorbit. The origin of the horizontal axis s is located at the midpoint between two hexapole segments.

will be slightly different and the voltage difference applied will not be identical. Theseeffects again lead to a deviation of the restoring force and hence to regions of instability.Note that the periodicity of these deviation is 1/Ωcycl rather than 1/nsegΩcycl. Experi-mental measurements of the stability of the synchrotron are presented in Sect. 3.4.

2.5 Longitudinal confinement

The previous section showed that molecules are confined transversally (in the y- andr-direction) by applying a constant voltage on the forty hexapoles that are placed ona circle. This section discusses how to confine a molecular packet in a tight bunchalong the longitudinal direction. The left side of Fig. 8 shows a schematic of the time-dependent switching scheme. The electric field is switched synchronously with therevolving packet in the synchrotron between the confinement field and the so-calledbunching field (the electric field of the bunching field is 4/3 times stronger than of theconfinement field). The time variation of the voltage in sync with the motion of themolecules is the eponym of the molecular synchrotron. On the right side of Fig. 8 theelectric fields for the confinement and the bunching configuration are shown as a func-tion of position along the equilibrium orbit (dashed green line on the left panel) in blackand red, respectively. For a time duration Δtbunch the electric field is switched to thebunching configuration every time the synchronous molecule is in the gap region. Thistime is translated into an effective bunching length Δxbunch, which is determined by thefinal velocity vf of the synchronous molecule that is injected by the Stark deceleratorbeam line

Δxbunch = vfΔtbunch . (24)

In Fig. 9 the operation principle of the longitudinal bunching scheme is describedin more detail. It is similar to that of the buncher in the decelerator beam line (describedin [49]). The upper panel shows the potential energy of the low field seeking componentof the |J, MK〉 = |1,−1〉 state as a function of longitudinal position along the equi-librium orbit for the two switching configurations. The confinement field corresponds




Fig. 9. The upper panel shows the potential energy of an ammonia molecule in the |J, MK〉 = |1, −1〉 stateas a function of longitudinal position along the equilibrium orbit. The potential of the confinement andbunching field is shown in black and red, respectively. The origin of the horizontal axis is located at themidpoint between two hexapole segments. In the gray shaded region the synchronous molecule experi-ences the bunching field. The effective length Δxbunch is 5 mm and is symmetric around the gap region.The middle panel shows the change in potential energy as a function of position. A molecule that is be-hind the synchronous molecule (left dotted line) experiences a negative change in potential energy and willbe pushed towards the synchronous molecule. The restoring force for a molecule is approximately linearfor a length of 5.0 mm. Near the synchronous molecule the slope of ΔW is 8.8 cm−1/m. The lower panelshows the phase-space diagram in the longitudinal direction. The solid lines correspond to molecules thatare stably bunched while trajectories in dashed lines are not stably bunched.

to the black and the bunching field to the red curve. Let us first consider the motionof the synchronous molecule along the equilibrium orbit in the confinement configura-tion. If the electric field is not switched, it will lose potential energy and gain kineticenergy as it moves along the electric fringe fields in the gap region. After passing themiddle of the gap, the synchronous molecule will regain all the potential energy it lostand will have the same kinetic energy as before it entered the gap. If now the elec-tric field is switched from the confinement field to the bunching field for an effectivelength Δxbunch symmetrically around the synchronous molecule (gray shaded region),the synchronous molecule will gain and lose a larger amount of kinetic energy. After theswitching process the velocity of the synchronous molecule still remains unchanged.

For non-synchronous molecules, the change in potential energy ΔW depends onthe longitudinal position s of the molecule along the equilibrium orbit as the field isswitched to the bunching configuration.

ΔW(s) = (Wbunch,a(s)− Wconf,a(s)

)− (

Wbunch,b(s +Δxbunch)− Wconf,b(s +Δxbunch))

. (25)

We will denote the position of the synchronous molecule when the field is switchedto the bunching and confining configuration by ±s0, respectively. Furthermore, weintroduce the Δs, the position of a non-synchronous molecule with respect to the syn-




chronous molecule when the field is switched to the bunching configuration, s = −s0 +Δs. The change in energy can now be written as

ΔW(s) = (Wbunch,a (−s0 +Δs)− Wconf,a (−s0 +Δs)

)− (

Wbunch,b (s0 +Δs)− Wconf,b (s0 +Δs))

. (26)

ΔW is shown in the middle panel of Fig. 9 for a synchronous velocity of 124.3 m/s,and an effective bunching length Δxbunch = 5 mm. The horizontal axis shows theposition of a molecule with respect to the synchronous molecule. The synchronousmolecule (Δs = 0) will not change its kinetic energy during the switching process.A molecule that is, e. g., – in position – behind the synchronous molecule (dotted line)experiences a stronger acceleration than deceleration in the gap and will gain kinetic en-ergy. Consequently, it will be pushed towards the synchronous molecule. Conversely,a molecule that is in front of the synchronous molecule will be decelerated and pushedback towards the synchronous molecule.

The average longitudinal force F that a molecule experiences while passinga hexapole segment of length ΔL can be expressed as

F = −ΔW(s)

ΔL. (27)

For molecules close to the synchronous molecule (Δs is small), the restoring force islinear and the force can be written as

F = −ΔW ′(s)Δs

ΔL, (28)

with ΔW ′(s) being the slope of the potential difference. ΔW ′(s)/ΔL is the harmonicforce constant k. The corresponding longitudinal angular frequency is

ωsyn =√

ΔW ′

mΔL. (29)

The green dashed line in the middle panel of Fig. 9 shows a linear fit of the potentialdifference ΔW . From this fit we find ΔW ′ = 8.8 cm−1/m. Furthermore, the length ofan hexapole segment ΔL = 2π(Rring + requi)/40 is 39.6 mm. Thus we find ωsyn/(2π) =58 Hz.

The lower panel of Fig. 9 shows a number of trajectories in phase-space calculatedusing the longitudinal potential (middle panel) and the resulting average force. In theharmonic part of the well (inner solid circle) the synchrotron frequency ωsyn/(2π) is60 Hz.

2.6 Numerical simulation

In order to get a better understanding of the synchrotron we have performed numeri-cal simulations of the trajectories of molecules inside the molecular synchrotron. Thebenefit of these simulations is that experimental measurements can be reproduced andconfirmed. The simulations also enable us to obtain quantities that cannot be directly




measured, and give us insight into the effects of misalignments and coupling of thedifferent motions in the synchrotron. Unfortunately, the coupling makes it difficult totest the program for errors in the code; due to the Coriolis coupling it is impossible toperform 1D calculations. In the next section some of the experimental measurementsare compared to simulations. This is not done for all measurements because a calcu-lation in which 100 000 molecules revolve in the molecular synchrotron for severalseconds is time consuming and requires a large amount of computer resources (CPUand RAM). For example, in the simulation of Fig. 17 (right panel) the trajectories of100 000 molecules are calculated for 100 000 different voltage-velocity configurationswith a Supercomputer using 100 CPUs. It took 4 1/2 days to finish the calculation. Thismakes it impractical to optimize the different initial conditions.

The symmetry of the molecular synchrotron implies that the electric potential ofonly one segment needs to be calculated precisely. The electric potentials for the othersegments are determined via a transformation of coordinates from the initial segment.The electric potential of the first segment is calculated as a grid with Simion [47] using20 grid points/mm. The force that a molecule experiences in this potential grid is de-termined by the second derivative of the potential field. In order to simulate a molecularbeam that is injected into the molecular synchrotron from a Stark decelerator beam line,a packet of molecules with a Gaussian position and velocity spread is generated. Thetrajectory of each molecule is traced until a certain number of round trips have beencompleted inside the synchrotron. If a molecule hits an electrode during its passage,or if it is outside of the hexapole, the calculation of that specific molecule is stoppedand a new trajectory with new initial conditions is started. The number of detectedmolecules at the end of the simulation corresponds to the relative density of moleculesin the experiment. A small additional random force is added to the confinement force ineach segment in order to reproduce the effect of small misalignments in the synchrotron(see Sect. 3).

The left three panels of Fig. 10 show the results of a simulation of the trajecto-ries of 100 000 molecules that make a 100 round trips within the synchrotron. The top,middle and bottom panels show the phase-space distribution in the radial, vertical andlongitudinal directions, respectively. The initial position and velocity distributions inthe three dimensions are 2.5 mm and 2.5 m/s (FWHM), respectively and are plotted asgray points in each panel. The initial conditions should be an adequate approximationto simulate the output of the two deceleration beam lines. The solid and dashed linesshow some trajectories in phase space resulting from the models presented in Sects. 2.3and 2.5. The phase-space distribution after 100 round trips is shown in green, red andblue for the radial, vertical and longitudinal direction, respectively. It is seen that theinitial phase-space distribution is smeared out in all six dimensions and fills the com-plete acceptance volume. The final distribution in the radial direction (green points)is displaced with respect to the model. The equilibrium radius is larger than that pre-dicted by the model (r ′

equi is 2.5 instead of 2.1 mm). This is attributed to the fact thatthe model assumes a hexapole that is bent into a perfect circle, while the numericalsimulations takes into account that the ring is formed by straight hexapoles. The fig-ures shows the radial distribution of the molecules in the middle of one segment. Itwould be more appropriate (but also more complicated) to use a time-averaged contourplot.




Fig. 10. For the radial, vertical and longitudinal directions, a phase-space plot is shown in the left top, mid-dle and bottom panel, respectively. In this simulation the trajectories of 100 000 molecules are calculatedwhile they make up to 100 round trips in the synchrotron. The gray points indicate the initial molecularphase-space distribution; the colored points show the final simulated phase-space distribution. The righttop, middle and bottom panel show the characteristic frequency as function of distance from the syn-chronous molecules from analytic models for the radial, vertical and longitudinal direction, respectively.

In the vertical direction the distribution (red points) does not fill the entire phase-space acceptance. The size of the vertical phase-space acceptance is determined by theamplitude of the vertical motion and its equilibrium orbit (see loss condition in Eq. (23)and substitute r ′ with r ′

equi). The vertical phase-space acceptance inherits the inaccuracyin r ′

equi. The longitudinal distribution (blue points) fills the entire phase-space accept-ance. The slight asymmetry in the longitudinal velocity is not yet understood.

The characteristic frequency for the radial, vertical and longitudinal directions asa function of distance to the synchronous particle is shown in the three right panels ofFig. 10. The characteristic frequency for the synchronous molecule is 902 Hz, 860 Hzand 58 Hz for the radial, vertical and longitudinal frequency, respectively. The valuesare similar or equal to the values listed in Table 1.

3. Experimental characterization of a forty segment synchrotron

In this section we will present the lay-out of the synchrotron. We give a detailed de-scription of the mechanical design and alignment procedure as well as the high voltageswitches and associated electronics. Packets of ammonia molecules with a tunable vel-ocity and adjustable velocity spread are injected clockwise and counter clockwise intothe synchrotron by two molecular beam decelerators. The first beam line was alreadyused in the experiments of Heiner and co-workers and is described in detail in ref. [49].The second beam line is newly built [15]. It is similar to the first beam line, differ-ing only in the number of stages that can be used for decelerating and bunching the




Fig. 11. Upper panel: A photograph of the molecular synchrotron consisting of 40 straight hexapoles sepa-rated by a 2 mm gap. Lower panel: A photo-realistic image of the molecular synchrotron together with thetwo Stark decelerator beam lines.

molecules. Furthermore, instead of a general valve, a Jordan valve is used. The twobeam lines give similar signals.

3.1 Experimental setup

A photograph of the assembled molecular synchrotron is shown in Fig. 11. The diam-eter of the ring is 500 mm. Each hexapole segment consists of six cylindrical highlypolished electrodes with a rod diameter of 4 mm, rounded off at each end. To guar-antee a constant gap of 2 mm between neighboring hexapole segments, the two innerelectrodes are 36.6 mm, the two middle electrodes are 37.4 mm and the two outer elec-trodes are 38.1 mm long (see Fig. 2). The electrodes are mounted on an aluminum oxidedisc that also serves as insulator. High voltage tests show that discharges between elec-trodes tend to occur along the shortest pathway across the ceramic surface. A slit inthe ceramic between two neighboring electrodes minimizes the chance of a discharge,by maximizing the path length along the ceramic. The ceramic disc together with theelectrodes is mounted with three screws onto a stainless steel holder. Using a specialalignment tool, the center of the hexapole is aligned with respect to the holder witha precision of less than a tenth of a millimeter. These holders are then placed with setpins onto an aluminum base plate under an angle of 9◦ with respect to each other. Theresulting precision between neighboring hexapole segments is on the order of one tenthof a millimeter.

The forty segments are internally connected in three groups: (i) 32 hexapoles(sketched in black in Fig. 2) are switched continuously between the confinement andbunching configuration. (ii) 4 hexapoles (sketched in green in Fig. 2), 2 hexapoles foreach beam line, are turned off to allow the injection of new molecular packet into themolecular synchrotron. (iii) 4 hexapole (sketched in red in Fig. 2), 2 hexapoles for each




beam line, are turned off to allow the injection of new molecular packet into the molecu-lar synchrotron and are switched to appropriate voltages to extract ions for detection.

3.1.1 High voltage switches

Seven high voltage power supplies are used for the two beam lines and the molecularsynchrotron (manufactured by FUG (HCN 700-12500) and Spellman (SL15*1200)).The necessary thirty high voltage switches (four for the first beam line, five forthe second beam line and twenty one for the molecular synchrotron) are manufac-tured by Behlke Electronic GmbH (HTS-151-03-GSM) and configured by the elec-tronic workshop of the Fritz-Haber-Institut. The bias voltage is generated by eightpower supplies (Delta Electronika BV (ES 0300-0.45)). The positive electrodes re-quire two switches in series to switch between the confinement, the bunching andthe detection/incoupling configuration. The negative electrodes switch between theconfinement and detection/incoupling configuration. The lower voltage input variesdepending on which electrode in which segment the switch is connected to. For themolecular synchrotron in total 21 high voltage switches are used: (i) 3 switches areused for the main section; one switch for the negative electrodes and two switches forthe positive electrodes, which switch between ground and −4 kV and between ground,+2 kV and +4 kV, respectively. (ii) 6 switches are used for the injection section; foreach beamline one switch for the negative electrodes and two switches for the positiveelectrodes, which switch between ground and −4 kV and between ground, +2 kV and+4 kV, respectively. (iii) 12 switches are use for the detection section; For each beamline, two switches are used to switch the voltage applied to the top electrodes between−300 V (detection), +2 kV (confinement) and +4 kV (bunching), one switch is usedfor switching the voltage applied to the middle top electrodes between −130 V (detec-tion) and −4 kV (confinement), two switches are used to switch the voltage applied tothe middle bottom electrodes between +130 V (detection), +2 kV (confinement) and+4 kV (bunching), and finally one switch is used to switch the voltage applied to thebottom electrodes between +300 V (detection) and −4 kV (confinement).

3.1.2 Detecting ammonia molecules with (2 + 1) REMPI

The packets that are injected by two Stark decelerator beam lines are detected simultan-eously using the same laser beam. The deuterated ammonia molecules are detected via(2+1) resonantly enhanced multi-photon ionization (REMPI) [50] using pulsed laserlight around 317 nm. To detect both beam lines simultaneously, the laser beam is re-flected and refocused ( f = 125 mm) inside the synchrotron into the detection point forbeam line 2. The ionized molecules are extracted by applying an appropriate voltage onthe detection hexapoles such that ions drift upwards between two hexapole segments(right side of Fig. 12). In a 50 cm long time-of-flight tube, kept on a potential of −1 kV,ions with different mass drift apart such that the deuterated ammonia molecules are sep-arated from ions of different mass [51]. These ions are detected on a microchannel plate(MCP). The measured ion signal is proportional to the ammonia density in the laserfocus.




Fig. 12. Detection Scheme. Left: A high power UV laser pulse (15 mJ/pulse – 5 ns duration) is generatedby pumping a dye laser with a Nd:YAG laser at 532 nm and doubling the resulting frequency with a BBOcrystal. The laser beam is focused in between two hexapoles of the synchrotron using a lens with a focallength of 500 mm. The molecules are ionized by a (2 +1) REMPI scheme. In this way molecules can bedetected at the injection point each time they have completed an integer number of round trips.Right: During the REMPI process extraction voltages are applied on the detection hexapoles. Positive ionsexperience an upward force and enter the time-of-flight tube. In the tube different masses drift apart andcan be measured selectively by a microchannel plate (MCP) detector.

3.1.3 Number of simultaneously stored packets

Theoretically a forty segment molecular synchrotron with forty gaps allows the sim-ultaneous confinement of eighty molecular packets: forty packets traveling clockwise,and forty traveling anti-clockwise. The molecular synchrotron is loaded such that eachstored packet is in a gap region when a new packet is injected. In practice the spacingbetween successive packets must be at least two segments as the detection and injectionhexapoles are connected in pairs of two. When a new packet is injected, the injectionand detection hexapoles need to be turned off. With a spacing of one segment, a packetin the same section of the synchrotron will therefore be lost. If only one beam line isused up to 20 packets can be stored simultaneously. If both beam lines are used theupper limit of stored packets from one beam line is determined by the injection anddetection technique of the other beam line. The procedure of injection and detection in-side the synchrotron is synchronized such that the stored packets are not affected. Withcounter-propagating packets in the ring this is best realized if all packets are locatedsimultaneously inside the main part of the first section consisting of the 24 hexapoles(see Fig. 2). The maximum number of simultaneously stored packets is

npackets = 2 ·(⌊

24

nspacing

⌋+1

), (30)

where nspacing is the spacing between successive packets in units of segments. Witha spacing of two hexapole segments, up to 26 packets can be confined simultaneously.In this paper the spacing between successive packets is generally two or three segments.To illustrate the advantage of confining multiple packets simultaneously, let us considera molecular packet with a velocity of 124.3 m/s. To measure a single packet after, e. g.,




1000 round trips and averaging each point 100 times takes 100 × 12.7 s = 21.2 min.With 13 simultaneous revolving packets, each measuring point takes only 1.8 min.

3.1.4 Continuous and pulsed trigger scheme

The detection laser and the deceleration beam lines require a repetition rate between 9and 11 Hz [52,53]. The time delay between the injection of successive packets, whichis referred to t100 ms, needs to be found such that

0.09 s < t100 ms < 0.11 s . (31)

The time delay t100 ms after a new packet is injected nspacing segments in front of the mostrecently injected packet is

t100 ms = (nrt ·40−nspacing

) · tseg . (32)

Here, nrt is an integer number and tseg the time that the synchronous molecule needsto pass one full segment. In this paper a packet makes 7 or 8 round trips before a newone is introduced. The desired trigger pulses are generated with arbitrary wave formgenerators (Agilent 33220A) and with modified delay clock generators (FHI electronicworkshop). A 10 MHz time reference clock is used such that all components have thesame time basis (TCXO standard).

In the measurements presented in this paper, two different trigger schemes are used:a pulsed and a continuous mode. In both modes multiple molecular packets revolveinside the synchrotron for the same number of round trips before being detected. De-pending on the experiment, one trigger scheme is advantageous over the other.

Pulsed trigger scheme. The upper panel of Fig. 13 shows a sketch of the pulsed triggerscheme. In this scheme, all timings are defined relative to a delay generator. This masterclock determines when and how many molecular packets are decelerated and injectedinto the molecular synchrotron. Each packet is decelerated and injected using a specifictime sequence. Between each sequence is a time delay of t100 ms such that the spacingbetween packets is exactly nspacing segments. Every packet stays in the synchrotron thedesired number of round trips before laser pulses are sent in to detect each one of them.The time delay between each of the laser pulses is again t100 ms. The advantage of thistrigger scheme is that an arbitrarily high number of round trips can be measured. Fora large number of packets the loading and detection time is rather long, making thisscheme less suitable for collision studies.

Continuous trigger scheme. The continuous trigger scheme is shown in the lower panelof Fig. 13. In this scheme, the molecular synchrotron switches continuously. Packets ofmolecules are injected with a delay time of t100 ms such that the spacing between packetsis exactly nspacing segments. The time at which the detection laser is triggered is rela-tive to the injection time. This allows the injection of a molecular packet into an emptygap after a previous molecular packet is detected and has exited the synchrotron. Withcounter-propagating packets present a molecular packet with a velocity of 124.3 m/sand a spacing of two segments can stay up to 103 round trips in the synchrotron before




Fig. 13. The two trigger schemes of the molecular synchrotron. Subsequent packets are injected and de-tected with a time delay t100 ms of about 100 ms to assure that the repetition rate of the pulsed dye laserremains between 9 and 11 Hz (see Eq. 32). The upper panel shows the pulsed triggering scheme for thesimultaneous confinement of eight packets. In this sketch we inject, e. g., eight molecular packets into thesynchrotron, then wait until each packet completes the desired number of round trips before eight laserpulses are fired to detect them. The switching between confinement and bunching configuration starts afterthe injection of the first packet. The lower panel shows the continuous scheme, in which the molecular syn-chrotron is operated continuously. After the time delay of t100 ms a new molecular packet is injected. Thetime of the laser pulse is changed relative to the injection time. For reasons of clarity the horizontal axisof the molecular synchrotron in the upper and lower panel is exaggerated.

a new packet takes it place (n is 8 and npackets is 13). The maximum number of roundtrips is given by

nrt,max = n ·npackets −1 . (33)

The disadvantage of the continuous scheme is the fixed maximum number of roundtrips. If, e. g., the depletion due to collisions is too small and it is desired to increasefurther the number of round trips it will be necessary to use the pulsed trigger scheme.

3.2 Packets of neutral molecules revolving for over a mile

Figure 14 shows the density of ammonia molecules as a function of time after injectioninto the synchrotron for a series of selected numbers of round trips. In this measurementthirteen packets are injected (npackets = 13) using the pulsed trigger scheme, after whichthe loading is stopped. These packets trail each other by a distance of three hexapoles(nspacing = 3). The first and the last packet are four hexapoles apart. The different peakscorrespond to characteristic number of round trips, e. g., for the 640th round trip the mo-lecular packet is confined for 8.15 s, corresponding to a flight length of one kilometer.Even after 1025 round trips, i. e., after the molecules have traveled a distance of overa mile and have passed through a gap 41 000 times, their signal can be clearly recog-nized. The temporal width of 21 μs corresponds to a spatial distribution of 2.6 mm.




Fig. 14. Measurements of the density of ND3. molecules as a function of time (in seconds) for a selectednumber of round trips. The observed temporal width of 21 μs after 1025 round trips corresponds to a lengthof 2.6 mm after a flight distance of more than a mile. The inset shows the exponential decay of the signalwith time. Figure is taken from reference [28].

This measurement explicitly demonstrates the stability of the trajectories of a molecularpacket inside a molecular synchrotron.

The density of ammonia molecules is seen to decay exponentially with time at a rateof 0.31 s−1 (see inset of Fig. 14). This is the lowest decay rate that has been observedfor neutral ammonia molecules in any trap to date. In all earlier electrostatic trappingexperiments, 1/e-lifetimes of only a small fraction of a second were observed, proba-bly limited by non-adiabatic transitions to non-trappable states near the trap center. Thiswas only realized when substantially longer lifetimes of up to 1.9 s were measured inan electrostatic trap that had a non-zero electric field at the center [4]. With the presentconfinement and bunching scheme, the molecules are never close to a zero electric fieldin the synchrotron. In addition, although the magnitude of the electric field is changed,its direction is not. This prevents the occurrence of non-adiabatic transitions inside thehexapoles [5]. A major contribution to the observed loss rate is optical pumping of theammonia molecules out of the |J, MK〉 = |1,−1〉 level by blackbody radiation, calcu-lated to occur at a rate of 0.14 s−1 at room-temperature [54]. The remaining loss-rateof 0.17 s−1 is well explained by collisions with background gas at the approximately5×10−9 mbar pressure in the vacuum chamber.

The longitudinal position spread can be determined from the time-of-flight (TOF)measurements presented in Fig. 14. Each of the TOF profiles is fitted with a Gaussianand from the fitted width the position spread is calculated. The spread is fairly constantover a trapping time of 13 s corresponding to 1025 round trips. The averaged positionspread is 2.9 mm.

In Fig. 15, the position spread of the 100th round trip is shown together with theposition spread resulting from numerical simulations. For the experimental data, thetime-of-flight is normalized and the peak maximum is set to zero seconds. To obtainthe position spread it is multiplied by the synchronous velocity (vf = 124.3 m/s). Thesimulation is performed using the input parameters as discussed in Sect. 2.6. For the




Fig. 15. The experimentally measured and simulated position spread after 100 round trips. The black curveis obtained by multiplying the measured time of flight profile shown in Fig. 14 by the velocity (124.3 m/s)of the stored molecules. The red curve is obtained by binning the data presented in Fig. 10.

simulated data, a normalized histogram of the molecules that made 100 round trips isshown as a function of the longitudinal position s.

3.3 Comparison between different bunching schemes

The bunching scheme described in Sect. 2.5 is different from the longitudinal confine-ment scheme employed in the two piece molecular synchrotron. To distinguish betweenthe two schemes the bunching of the previous molecular synchrotron is referred toas ‘old scheme’ and that of the current molecular synchrotron as ‘new scheme’. Inthe old two segment molecular synchrotron, every time the molecular packet entersa gap region the electric field is switched between different configurations to achievea longitudinal confinement force. The old bunching scheme is described in detail inreferences [24,27]. Briefly, when the synchronous molecule is near a gap between ad-jacent hexapoles, the voltages on the hexapole in which it currently resides (hexapole1) are switched to ground while the voltages on the next hexapole (hexapole 2) areswitched such as to generate a strong inhomogeneous electric field. As a result of thisthe molecules are decelerated when they leave the first hexapole. When the synchronousmolecule is exactly between the two hexapoles, the voltages on the second hexapoleare switched to ground while the voltages on the first hexapole are switched such asto generate a strong inhomogeneous electric field. As a result of this the moleculesare accelerated when they enter the second hexapole. The synchronous molecule is ac-celerated as much as it is decelerated and its kinetic energy remains unchanged. Incontrast, molecules that are in front of the synchronous molecule experience a net de-celeration while molecules that are behind the synchronous molecule experience a netacceleration. The molecular packet is ‘bunched’ longitudinally. To compensate for thelack of transverse focusing in the gap, the confinement focusing force is increased fora short time period before and after the gap region. Compared to the bunching schemedescribed in Sect. 2.5, the old bunching scheme results in an effective longitudinal po-tential well that is ten times steeper.




Fig. 16. Left panel: Measurements of the density of ND3. molecules as a function of time inside the 40segment molecular synchrotron for the two different bunching schemes (logarithmic scale). For both meas-urements the molecular packet is bunched every fifth hexapole segment (8 times per round trip). The blackpoints refer to the bunching scheme that was used in the previous molecular synchrotron consisting of twohalf rings (‘old scheme’) [24]; the red points correspond to the bunching scheme which is described inSection 2.5 (‘new scheme’). The lifetime of the molecules in the synchrotron is determined from the expo-nential fit. Right panel: The ammonia decay measurement of the old bunching scheme for three differentnumbers of bunching per round trip; in black the molecular packet is bunched 8 times, in blue 4 times andin green 2 times per round trip.

To compare both switching schemes the wiring of the synchrotron is changed.Five hexapole segments are wired together (yielding 8-fold symmetry) and only onemolecular packet is stored at a time. The left panel of Fig. 16 plots the densityof the molecular packet for both bunching schemes on a logarithmic scale togetherwith a fitted exponential decay. The new scheme (red circles) results in a lifetimeof 1.4 s while the old scheme (black squares) results in a lifetime of 54 ms. To un-derstand the fast exponential decay, the right panel of Fig. 16 shows the exponen-tial decay of the old scheme when the bunching procedure is applied 2 (green datapoints), 4 (blue data points) and 8 (black data points) times per round trip. Whenthe number of bunching sequences per round trip is increased, the experimental de-cay becomes faster. The loss may be attributed to two different loss processes: (i)The molecules undergo a spin-flip transition (non-adiabatic transition) to a state thatis not low-field seeking while one hexapole is grounded. (ii) Molecules are lost dueto the lack of transverse focusing while the molecules are being bunched. In the ex-periments with the 2 segment molecular synchrotron the losses were dominated byother loss mechanisms. For the forty segment molecular synchrotron, on the otherhand, the old bunching scheme leads to unacceptably large losses. The stability ofthe new bunching scheme well outweighs the lower acceptance of the new scheme.In addition, the new bunching scheme is less demanding for the high voltage elec-tronics; in principle only one high voltage switch is required. The need to injectand detect multiple packets increases the number of required switches strongly (seeSect. 3.1.1).




Fig. 17. Experimental and simulated ammonia intensity after 100 round trips as a function of velocity andvoltage. The left panel shows the experimental data. The ammonia density is determined at 200 differ-ent voltages and 39 different velocities (200×39 grid), averaging 52 times at each point. The right panelshows the numerically calculated intensity at 100 different velocities and 100 different voltages, resultingfrom simulating trajectories of 100 000 molecules as they make up to 100 round trips in the synchrotron.The solid curves that overlay the contour plots indicate when the horizontal (red curve) or vertical (whitecurve) tune is an integer value. The tunes are calculated using Eq. (18) and the angular frequency derivedfrom an analytic function of the electric field [38].

3.4 Transverse motion – stopbands

Due to imperfect alignment of the hexapoles, the confinement force varies from onesegment to the next. In a ring structure this periodic variation causes motional res-onances. As a result, certain forward velocities cannot be confined stably at a givenhexapole voltage (see Sect. 2.4). In this section we study these so-called stopbands ex-perimentally. Figure 17 show the ammonia intensity after 100 round trips. The left panelshows experimental data while the right panel shows the results of a numerical simula-tion of the experiment. The voltage is scanned between 3 and 7.5 kV in 200 steps whilethe velocity is scanned between 100 and 138 m/s in 39 steps. At each setting the signalis averaged over 52 shots. In the simulation, the trajectories of 100 000 molecules arecalculated at a 100 different velocities and 100 different voltages.

A number of features of Fig. 17 deserve special attention:(i) The observed ND3 intensity is largest for high velocities simply because the

beamline delivers a more intense beam (see [49]).(ii) When the velocity is large and the voltage is relatively low (top left cor-

ner in Fig. 17) the equilibrium orbit lies outside the hexapoles and no molecules areconfined.

(iii) When the velocity is small and the voltage is high (bottom right parts in Fig. 17)the observed intensity is small. Under these conditions the equilibrium radius is smalland the molecules spend considerable time at the geometrical center of the hexapoleswhere the electric field is small. We attribute the observed losses to Majorana tran-sitions [4,5] that transfer molecules to the untrapped |J, MK〉 = |1, 0〉 state. This is




implemented in the simulation by assuming molecules to be lost once they experiencea Stark shift below 0.015 cm−1. We chose this value to get the best agreement with theexperimental data. This frequency is generated by the rapid switching (100 ns) of theelectric fields.

(iv) As a consequence of (ii) and (iii), the highest intensity is observed when the vel-ocity and voltage are such that the equilibrium radius is around 2 mm. Note that in anideal hexapole field when the inversion splitting is neglected, the equilibrium radius isproportional to vf/

√V0.

(v) At certain combinations of the velocity and voltage no signal is observed. Theseso-called ‘stop bands’ are the result of motional resonances when the betatron tune, thenumber of transverse oscillation per round trip, takes on an integer value. The white andred curves that overlay the contour plot indicate integer values for the radial or verticalbetatron tune. The occurrence of stopbands implies that the alignment of the individ-ual hexapoles is not perfect, or the voltage applied to the hexapoles is not equal for allhexapoles. In order to reproduce the measurements, we introduced random misalign-ments in the numerical simulations. This is done by calculating six random numbersfor each segment; three of these numbers are used to shift the center of the hexapolein three directions up to 20 μm, while the other three are used to scale the magnitudeof the force by up to ±2%. It is seen, that although the simulation looks very similarto the measurements, the simulated stopbands are shifted to slightly higher velocities.A possible cause for this might be that the numerically calculated electric field used inthe simulations is slightly smaller than the true electric field.

From the measurements of the stopbands and with the help of Eq. (19), we can de-termine the horizontal and vertical betatron frequencies. From Fig. 17, we see that ata voltage difference of 6 kV, the radial and vertical tune are equal to 11 at 126.9 m/s and122.5 m/s, respectively. This corresponds to a radial betatron frequency of 902 Hz anda vertical betatron frequency of 851 Hz. Additionally, the radial frequency can be deter-mined when the radial tune is equal to 12, which corresponds to a velocity of 119 m/sat 6 kV. This second measurement of the radial frequency corresponds to 881 Hz. Theresulting average radial frequency is 891 Hz with an error of 11 Hz.

3.5 Longitudinal motion

Non-synchronous molecules are trapped in an effective potential that moves along withthe synchronous molecule. Close to the synchronous molecule the well is harmonic andnon-synchronous molecules will oscillate around the synchronous molecules at the syn-chrotron frequency, ωsyn. In this section, we will determine the synchrotron frequencyby measuring how quickly a stored packet expands if the longitudinal confinement forceis switched off – i. e., if constant voltages are applied to the synchrotron.

The spatial distribution of the molecules as a function of time is given by

Δs(t) =√

(Δsi)2 + (Δvs · (t − tmin))

2. (34)

Here, Δsi is the minimum position spread and Δvs is the velocity spread of the packet.tmin is the time at which the molecular packet has a minimal position spread. This time isnot equal to the last time the fields are being switched but slightly later, as the molecules




Fig. 18. The longitudinal expansion of an ND3 packet in the molecular synchrotron; the left panel showsthe position spread of the molecular packet after 5 round trips (RT) as a function of the expansion time.The black and red points correspond to a situation when 2 and 40 bunching stages are used per round trip,respectively. The solid lines show the fit performed using Eq. (34). The two right panels illustrate the ex-pansion; after the second round trip the electric field is kept constant and the packet spreads out. The upperand lower right panels show the expansion with 40 and 10 bunching sequences per round trip. The solidwhite line shows the theoretical expansion using Eq. (34) with ωsyn/(2π) taking a value of 58 and 29 Hzfor the upper and lower right panel, respectively. To achieve the best contrast for the two dimensional plots,each time-of-flight trace (horizontal axis) is normalized separately.

are first focused in free flight before they start to expand. If, e. g., the molecular packetis bunched 40 times per round trip before they are released, tmin is (1/2) · (Lseg/vs) afterthe time the fields are switched for the last time. When the molecular packet is bunchedonly 2 times per round trip before release, tmin is 10Lseg/vs after the time that the fieldsare switched for the last time. Once the position spread and velocity spread of the packetis determined, the synchrotron frequency is found via

ωsyn = Δvs

Δs. (35)

The panels on the right hand side of Fig. 18 show the ammonia intensity as a func-tion of arrival time relative to the synchronous molecule (x-axis) for making a variablenumber of round trips after being released (y-axis). The molecular packet is confinedfor two round trips (RT), after which the switching of the electric field is stopped. Withconstant voltages, the molecules are confined transversally but are free to spread out inthe longitudinal direction. In the upper and lower right panels the molecular packet isbunched 40 and 10 times per round trip, respectively.

The panel on the left hand of Fig. 18 shows the longitudinal position spread ofthe molecular packet obtained from the time of flight data as a function of time afterrelease. In this case the molecules have been stored for 5 round trips (RT) before re-lease. The black and red squares correspond to measurements where the molecular




Fig. 19. Time-of-flight measurement showing 19 packets of ammonia molecules with a forward velocity of120 m/s revolving simultaneously in the molecular synchrotron. The horizontal axis shows the time rela-tive to the time at which the first injected packet has completed 150 round trips. Each packet is labeledby two numbers: the main number labels the order of injection whereas the subscript labels the number ofcompleted round trips. Figure reproduced from [28].

packet is bunched 2 and 40 times per round trip, respectively. The solid line showsthe fit to the data using Eq. (34). If the packet is bunched 40 times per round trip (redcurve), the fitted initial position and velocity spread are 2.9 mm and 1.2 m/s, respec-tively. Using Eq. (35), this corresponds to a synchrotron frequency, ωsyn/(2π) of 68 Hz.As discussed in Sect. 2.5 the synchrotron frequency becomes smaller when the lengthbetween bunching stages is increased

ωsyn ∝ 1√L

, (36)

where L is the period length. If the bunching field is switched only twice per round trip(black curve), L is 20 times larger. The frequency should be a factor of 4.5 smaller,around 15 Hz. Indeed measurements taken when the bunching field is switched twiceper round trip give a position spread of 5.5 mm and a velocity spread of 0.5 m/s (leftpanel), yielding a frequency of 14 Hz.

3.6 Multiple packets

Figure 19 shows a measurement of nineteen co-propagating packets of ammoniamolecules with a forward velocity of 120 m/s, loaded from beam line 1. The horizontalaxis shows the time relative to the time at which the first injected packet has com-pleted 150 round trips. The last injected packet has been in the ring for 320 ms and hasmade 24 round trips while by then the first injected packet has already been stored formore than 2 s. The loading scheme is such that each packet completes 7 round tripsplus the length of 2 hexapoles before the next packet is injected. In the measurementthe packets are seen to trail each other by 666 μs – precisely the time it takes them to




fly through two hexapoles. The time period between the first and last injected packetis equal to an effective length of four hexapole segments. The injection of multiplecounter-propagating packets will be discussed in Sect. 4.

4. Towards observing collisionsIn collision studies involving counter-propagating packets of molecules, losses in thenumber of stored molecules due to (in)elastic collisions accumulate over the number ofinteractions. Let us consider n packets of molecules revolving clockwise and n pack-ets revolving anti-clockwise, all at the same speed. After m round trips, each packetwill have interacted 2 ·n ·m times with counter-propagating packets. In this section thecurrent status of the state-of-the-art molecular synchrotron is discussed. We present ex-periments in which 26 packets of ND3 molecules are simultaneously confined in thesynchrotron; 13 packets revolving clockwise and 13 packets revolving anti-clockwise.The sensitivity to a collision after 100 round trips will be about 103 times increased. Un-fortunately, the sensitivity of the measurements conducted has so far been insufficient todetect the small additional losses in signal brought about by collisions.

4.1 Collisions between counter-propagating packets

Figure 20 shows a schematic of the synchrotron that illustrates the confinement of 26packets. The 13 orange circles indicate packets from decelerator 1 which revolve anti-clockwise. The 13 packets from decelerator 2 propagate clockwise and are shown asred circles. The distance between neighboring packets is two hexapole segments whilethe distance between the first and the last injected packet is 16 hexapole segments.A time-of-flight measurement that corresponds to this situation is shown in Fig. 21. Themolecules have a forward velocity of 124.3 m/s. The blue curve shows the ammoniaintensity measured with the detector in beam line 1, while the green curve shows theammonia intensity measured with the detector in beam line 2. t = 0 ms corresponds tothe time when a new packet is injected. The label above the measurements indicates thenumber of round trips a particular packet is confined in the synchrotron before it is de-tected (main number) and which beam line is used to inject it (subscript). The observedtime of flight consists of a series of peaks with a periodicity of 12.7 ms – correspondingto the time the synchronous molecules require to complete one round trip. Let us focuson the time of flight measured with the detector in beam line 1 (blue curve). As can beseen from the labeling of the peaks, the clockwise and anti-clockwise packets partiallyoverlap each other when being detected. The first five peaks and the last five peaks cor-respond to packets that are injected by beam line 1 and 2, respectively. The eight peaksin the middle of the series contain contributions from both beam lines. The time dif-ference between adjacent peaks is 636 μs – the time it takes the molecules to pass twosegments. The time difference between the last detected packet of beam line 2 and thefirst detected packet of beam line 1 corresponds to a spacing of 4 segments. If a newinjected packet traveled ten segments in the molecular synchrotron, when it reaches thedetection zone, it overlaps with the packet from the other beam line that is confined for40 round trips. This effect can be seen around 3.2 ms, where the ND3 intensity due tothe packet from beam line 2 increases significantly.




Fig. 20. A photo realistic image of the molecular synchrotron to illustrate the confinement of 26 packets.13 packets from decelerator 1 propagate counter clockwise (orange circles) and 13 packets from deceler-ator 2 propagate clockwise (red circles). The distance between neighboring clockwise or anti-clockwiserevolving packets is two hexapole segments. The first and the last packet are separated by 16 hexapole seg-ments. At two points in the ring the molecular packets are both injected and detected. The dashed blue andgreen arrows indicate the laser beam which ionizes the molecules at the first and second detection zone,respectively.

Fig. 21. A time-of-flight measurement of ammonia molecules in the molecular synchrotron when bothbeam lines are triggered via the continuous trigger scheme (discussed in Section 3.1.4). The signalrecorded at detectors 1 and 2 is shown in blue and green, respectively. t = 0 ms corresponds to the timewhen a new molecular packet is injected. The peaks that are observed correspond to 26 (2×13) molecu-lar packets and are labeled by two numbers. The main number indicates how many round trips the packetrevolved in the molecular synchrotron before being detected; the subscript indicates the beam line that isused for its injection. The color labeling follows Fig. 20.




Measurement scheme of a molecular collider. After successfully confining multiplecounter-propagating packets, the next step is to use the molecular synchrotron as a mo-lecular collider. We will first present a mathematical description of the experiment.Let us analyze how the ammonia intensity changes as a function of time if counter-propagating packets are present. If a molecular packet of ND3 molecules interacts withanother packet of ammonia over a length of dx, the change in the ammonia density canbe expressed as

d [ND3] = − [ND3] ·dx ·σtot · [ND3] . (37)

Here, we assume that the density of the two interacting packets is the same. Introducingthe relative velocity

vrel = dx

dt(38)

the change of the ammonia density can be rewritten as

d [ND3]

dt= −kND3 [ND3]2 . (39)

Here, kND3 is the decay rate due to bi-molecular collisions. Separating the variables andintegrating both sides the density of the ammonia molecules as a function of time canbe written as

[ND3] (t) = [ND3]0

1+ kND3 · t · [ND3]0

. (40)

To understand the difficulty of colliding two Stark-decelerated molecules, let us cal-culate the probability of a collision taking place between an ammonia moleculewith a molecule from a Stark-decelerated beam of ammonia with a density of1×106 molecules/cm3 and a length of 2 mm. If we assume a total cross sectionfor ND3-ND3 collisions of 500 Å2, the probability of a collision is P = σtot · dx ·[ND3] = 10−8. As described above, the molecular synchrotron can enhance this sig-nal by a factor of 103, such that the observation of such a collision in the molecularsynchrotron becomes more realistic. Because of the low probability of a collision, theexpected change of the decay rate due to bi-molecular collisions in the molecular syn-chrotron is expected to be small. In order to detect this extra loss, the synchrotron istoggled between three different modes.

The ammonia density is measured without counter-propagating packets (WO)The ammonia density is measured with counter-propagating packets (W)The background ND3 in the vacuum chamber is determined (BG)

The measurement without counter-propagating packets is used as a reference to includethe effects of the blackbody radiation and the collisions with the background gas. Thenormalized collision signal is

Ctotal = (WO−BG)− (W−BG)

(WO−BG). (41)




Fig. 22. The trigger scheme used for studying collisions between counter-propagating packets. To meas-ure the collisions as sensitively as possible, the synchrotron is switched between three modes: beam lines1 and 2, only beam line 1 or only beam line 2 inject packets into the synchrotron. Additionally, the as-signment for each beam line to the collision signal (W, WO and BG) is shown. If a molecule is detectedin the W mode, counter-propagating packets are present. WO stands for ‘without’ counter-propagatingpackets. In the background measurement (BG) no molecules from this beam line are injected; only counter-propagating packets revolve in the synchrotron. The time between each mode is a integer multiple of t100 ms.

Fig. 23. An example measurement to determine collisions between counter-propagating packets of ND3

molecules. The top panel shows a 100 000 single shot measurement in which a packet of molecules trav-eled 64 1

4round trips while the trigger scheme described in Fig. 22 is applied. The lower panel shows

a zoom-in of the of the panel above. The switching trigger scheme is directly visible.

If Ctotal is clearly greater than zero, this is an indication that bi-molecular collisions be-tween counter-propagating packets occur. From the increase of Ctotal as a function of thenumber of round trips, a bi-molecular collision rate can be determined.

The main advantage of this toggle mode is that slow drifts of the ammonia density,due to e. g. temperature changes in the valve or variation of the laser power, are compen-sated. The toggle scheme is sketched in Fig. 22. It shows as a function of time when thebeam lines inject packets of molecules. In one mode only beam line 1 operates, in the




Fig. 24. The averaged ND3 signal as a function of measurement cycles using the data of Fig. 23. The topand the bottom panel show the analyzed result for beam line 1 and beam line 2, respectively. In each panelthe data with collisions, without collisions and with background are shown in red, black and green, respec-tively. The dotted lines show the corresponding statistical error (1 ·σ).

Table 2. Result of the statistical analysis of Fig. 24 for the two beam lines.

beam line 1 beam line 2

W 2.15±0.22 0.83±0.12WO 2.19±0.22 0.73±0.13BG 0.01±0.02 0.01±0.02

second mode only beam line 2 injects packets, while in the last mode the synchrotronconfines clockwise and anti-clockwise revolving packets simultaneously. Every timethe synchrotron is switched between different modes 1.3 s are required to load all pack-ets. The detected molecules in this time period cannot be used for the collision analysis.To assure the continuous operation of the synchrotron the time duration of a mode isa multiple of t100 ms. This multiple is chosen such that the ratio of the loading time andthe time of one mode is minimal, while also keeping the time of one mode sufficientlyshort that drifts in signal level on relatively short time scales are compensated.

A first collision test. Analysis of the collision signal is only possible if the ND3 inten-sity is assigned correctly to one of the two injection beam lines. At certain numbersof round trips this is not possible; particularly when the counter-propagating packetsoverlap in the detection zone. For example the 40th round trip cannot be used, becauseit overlaps with the quarter round trip from the other beam line. For this reason, onlyround trips 0 to 39 and 64 1

4to 103 1

4can be used for the collision analysis. In the first

case the molecules are detected at the same point where they are injected. In the lattercase, the detection point is a quarter round trip further than the injection point.




Figure 23 shows a toggle measurement in which ammonia molecules are made torevolve 64 1

4round trips in the molecular synchrotron. The upper panel shows the ND3

intensity measured with detector 1 (blue curve) and detector 2 (green) for 100 000 sin-gle measurements, requiring a total measurement time of 2.8 h. The signal-to-noiseratio obtained with detector 1 is 1.2 : 1, while the one measured with detector 2 isslightly smaller. The lower panel shows an inset of the two data traces in which thetoggle mode in clearly visible.

The result of the analysis of the data shown in Fig. 23 is shown in the upper andlower panel of Fig. 24 for beam line 1 and 2, respectively. The horizontal axis indi-cates the number of cycles while the vertical axis shows the averaged ND3 signal forthe three different modes; the averaged ammonia signal with (black) and without col-lisions (red), and the background (green). In the analysis the first nine points of eachcycle for which the loading of the molecular synchrotron is not yet completed are neg-lected. The total averaged signal and the corresponding variances over all cycles isshown for each trace as a dashed and dotted line, respectively. Additionally they arealso listed in Table 2. As can be seen, the reduction in signal arising from bi-molecularcollisions is currently smaller than the statistical error associated with the experimentalmeasurements.

5. Conclusion

In this paper, we present the third generation molecular storage ring consisting of fortystraight hexapoles. The new mechanical design as well as the new electronic schemeused for switching the voltages results in a very stable trap for polar molecules. Thestability is illustrated by the fact that the density of the stored ammonia moleculesdecays at a rate of 0.31 s−1; the lowest decay rate that has been observed for neutralammonia molecules in any trap to date. This has made it possible to observe packetsof molecules after they have completed 1000 round trips in the synchrotron, therebytraveling a distance of over one mile. We present measurements that characterize theforces experienced by the molecules inside the synchrotron. A comparison betweenthese measurements and simulations show that we have a good understanding of thesynchrotron.

An important motivation of our work is the possible use of the synchrotron as a low-energy collider for cold molecules. In Sect. 4 measurements are shown that demonstratethat 26 packets of ND3 molecules can be stored simultaneously in the synchrotron;13 packets revolving clockwise and 13 packets revolving anti-clockwise for up to 100round trips. This results in a 103 fold increase in the number of collision events thattake place in comparison to a crossed beam setup. Unfortunately, so far we have beenunable to detect collisions between counter propagating packets. In order to be ableto detect collisions in the future, we are considering possible improvements to thesetup.

Every improvement that increases the lifetime of a molecular packet automaticallyimproves the signal-to-noise ratio. One obvious way to increase the lifetime is to coolthe molecular synchrotron to liquid nitrogen temperature. By this, the loss due to theblackbody radiation is reduced by a factor of 22; the decay rate of the blackbody




process drops from 0.14 s−1 at room temperature to 0.0063 s−1 at 77 K [54]. A bene-ficial side effect is that the cooling leads to a lower pressure in the vacuum chamber.A device that shields the molecules from the blackbody radiation would at the sametime function as a cold finger. Background gas will stick on the cold surface, de-creasing the density of the background gas. The lifetime of the molecules confined inthe synchrotron would increase significantly. Although this idea is straightforward, itsimplementation is not. It is mechanically and electronically highly demanding to op-erate a cooled molecular synchrotron. The ground potential of a cooling shield wouldbe only a few millimeters away from electrodes or wires that are on high voltage.This may cause discharges that might make it impossible to operate the molecularsynchrotron.

Another way to improve the sensitivity would be to implement velocity map imag-ing (VMI) [56]. With a suitable lens geometry to compensate for the astigmatic ex-traction fields [57], it should be possible to have a velocity resolution that is sufficientto separately detect the clockwise and anti-clockwise revolving packets. This leads toa significantly increased signal-to-noise ratio.

A completely different approach to measuring collisions would be to inject an (un-decelerated) molecular beam co-propagating with the stored packets. The molecularvalve that delivers the ‘target’ molecules can be synchronized with the cyclotron fre-quency, such that a stored packet of molecules will encounter a fresh target everyround trip. The advantage of this method is that it offers more freedom in choosingthe collision partners. For instance, in this way collisions between ammonia and hy-drogen molecules or hydrogen atoms could be studied. The disadvantage is that thenumber of collisions now scales only linearly with the number of round trips ratherthan quadratically, however the greater intensity of the target beam should make up forthis.

In conclusion, the molecular synchrotron presented here offers unique possibilitiesfor collision and other types of molecular physics studies. We have completely charac-terized the motion of the molecules in the ring and are ready to use the synchrotron aslow energy collider.

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A Forty-Segment Molecular Synchrotron

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